= Solution
For the <character of a localized orbital basis>, take the <trace> of the transformation <matrix>: both permutation of centres and signs of <atomic orbitals> matter. The two metal centres lie on the principal axis and remain fixed under rotation about $z$. At either metal the $(p_x,p_y)$ <basis> has the <matrix>
$$
R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.
$$
The full four-dimensional <group representation> is block diagonal with two such blocks. Consequently $\chi(R)=4\cos\theta$, and for the $60^\circ$ rotation of the <D6h molecular point group>, \b[$\chi(C_6)=4\cos(\pi/3)=2$].
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